Abstract

We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping is a self-derivation at each step. Following up on this idea we obtain several results and conclusions of interest. We also discuss the notion of a couplet on , consisting of a two-step derivation and its square , for example, whose defining property leads to further observations on the underlying ranked trigroupoids also.

1. Introduction

The notion of derivations arising in analytic theory is extremely helpful in exploring the structures and properties of algebraic systems. Several authors [1, 2] studied derivations in rings and near rings. Jun and Xin [3] applied the notion of derivation in ring and near ring theory to -algebras. In [4], the concept of derivation for lattices was introduced and some of its properties are investigated. For more details, the reader is referred to [3, 5–7].

Iséki and Tanaka introduced two classes of abstract algebras: -algebras and -algebras [8, 9]. Neggers and Kim introduced the notion of -algebras which is another useful generalization of -algebras and then investigated several relations between -algebras and -algebras as well as several other relations between -algebras and oriented digraphs [10]. Kim and Neggers [11] introduced the notion of and obtained a semigroup structure. Bell and Kappe [1] studied rings in which derivations satisfy certain algebraic conditions. Alshehri [12] applied the notion of derivations in incline algebras.

The present authors [13] introduced the notion of ranked bigroupoids and discussed -self-(co)derivations. In addition, they defined rankomorphisms and -scalars for ranked bigroupoids and obtained some properties of these as well. Recently, Jun et al. [14] obtained further results on derivations of ranked bigroupoids, and Jun et al. [15] introduced the notion of generalized coderivations in ranked bigroupoids and showed that new generalized coderivations of ranked bigroupoids are obtained by combining a generalized self-coderivation with a rankomorphism.

In this paper, we extend the theory of derivations on a ranked bigroupoid to that of a type of derivation on ranked trigroupoids, that is, two-step derivations on ranked trigroupoids considered as a couple of ranked bigroupoids and with such a two-step derivation on if it is a self-derivation on both and . The role of the operation in this definition is the more interesting one since it acts as the minor operation in and the major operation in . From the results obtained below it is clear that it is indeed possible to obtain meaningful insights, especially via the notion of a couplet on a ranked trigroupoid consisting of a pair of mappings , satisfying a natural condition (6) stated below which arises in a rather natural way from the context and is seen to be of interest in this study and presumably of any followup as well.

2. Preliminaries

An -algebra [10] is a nonempty set with a constant and a binary operation “” satisfying the following axioms:(A),(B),(C) and imply for all .

A -algebra is a -algebra satisfying the following additional axioms:(D),(E) for all .

Given a nonempty set , we let denote the collection of all groupoids , where is a map and where is written in the usual product form. Given elements and of , define a product “” on these groupoids as follows:
where
for any . Using that notion, Kim and Neggers proved the following theorem.

Theorem 1 (see [11]). is a semigroup; that is, the operation “” as defined in general is associative. Furthermore, the left-zero-semigroup is the identity for this operation.

A ranked bigroupoid is an algebraic system where is a nonempty set and “” and “” are binary operations defined on . We may consider the first binary operation as the major operation, and the second binary operation as the minor operation.

Example 2 (see [16]). A -algebra is defined as an algebraic system where is a group and where . Hence every -algebra is a ranked bigroupoid.

Example 3 (see [13]). We construct a ranked bigroupoid from any -algebra. In fact, given a -algebra , if we define a binary operation “” on by for any , then is a ranked bigroupoid.We introduce the notion of “ranked bigroupoids” to distinguish two bigroupoids and . Even though in the sense of bigroupoids, the same is not true in the sense of ranked bigroupoids. This is analogous to the situation for sets, where but in general.Given an element , has a natural associated ranked bigroupoid ; that is, the major operation and the minor operation coincide.Given a ranked bigroupoid , a map is called an -self-derivation if for all ,
In the same setting, a map is called an -self-coderivation if for all ,
Note that if is a commutative groupoid, then -self-derivations are -self-coderivations. A map is called an abelian--self-derivation if it is both an -self-derivation and an -self-coderivation.

3. Two-Step Derivations and Couplets on Trigroupoids

An algebraic system is said to be a ranked trigroupoid if algebraic systems and are ranked bigroupoids. A two-step derivation on a ranked trigroupoid is a mapping such that is both an -self-derivation and an -self-derivation.

Obviously, if one considers ranked -groupoids , then one may consider -step derivations for which one has as an -self-derivation for .

In this paper we will mostly be interested in the case of two-step-derivations on ranked trigroupoids and some related pairs of maps which we call couplets. For ranked -groupoids where , we obtain triplets, quadruplets, and so forth, as the appropriate generalizations.

Let be a two-step derivation on a ranked trigroupoid . Then, for any , we have

If we let , then it follows that

We call a couplet on a ranked trigroupoid if it satisfies condition (6), and if contains a constant 0.

Example 4. Let be the set of all real numbers and let be the ranked trigroupoid where and are usual multiplication, addition, and subtraction, respectively. If we let be a couplet on the ranked trigroupoid , then
If we let in (7), then
Thus for all , whence for all . If we let in (7), then . It follows that, for any ,
which implies that . Hence, by (7), we have ; that is,
If we let and in (9), then we have . Hence for all .

In Example 4, if is a two-step derivation on , then and for any . It follows that , which proves that for all .

Example 5. Let and let be a ranked trigroupoid where , + are usual multiplication and addition, respectively. Define a map satisfying for all . For , , , we define a map by where and . Then is a -self-derivation. In fact, if , for some , then
Assume that is a couplet on the ranked trigroupoid for some self--derivation . Then,
Since , we obtain

Proposition 6. There is no two-step derivation on the ranked trigroupoid .

Proof. Assume that is a two-step derivation on . Then, for any ,
If we let in (15), then
If we let in (16), then . On the other hand, if we let and in (15), then , proving that is a contradiction. Hence there is no two-step derivation on the ranked trigroupoid .

Proposition 7. There is no two-step derivation on the ranked trigroupoid .

Proof. Assume that is a two-step derivation on . Then, for any ,
If we let in (19), then and hence for all . If we let in (18), then , for any , is a contradiction.

Proposition 8. Let be a couplet on a ranked trigroupoid . If and , then for all . In particular, if , then is a -self-derivation.

Proof. If is a couplet on a ranked trigroupoid , then, for any , we have
Since , if we let in (20), then . It follows that
Hence for any . If we change into for any , then we obtain
for all . In particular, if , then and hence ; that is, is a -self-derivation.

4. Frame Algebras and -Algebras

A groupoid is said to be a frame algebra if it satisfies the axioms , and(F),

for all .

Example 9. Every -algebra is a frame algebra. Every lattice implication algebra (see [17, 18]) is a frame algebra.The collection of frame algebras includes the collection of -algebras and it is a variety. In a frame algebra the element 0 is unique. Indeed, if and are both zeros, then , yields .

Proposition 10. The collection of all frame algebras forms a subsemigroup of the semigroup .

Proof. Given frame algebras , if we let , then for all . It follows that , , and , proving that is a frame algebra. This proves the proposition.

Given groupoids , we define

Proposition 11. Let and be frame algebras. If , then is a frame algebra.

Proof. If , then for all . It follows that implies that . Moreover, shows that , and shows that , proving that is a frame algebra.

A ranked trigroupoid is called an fr()-algebra if(G) are frame algebras,(H).

Theorem 12. Let be an fr-algebra. If is a two-step derivation on , then (i),(ii) for all .

Proof. (i) If is a two-step derivation on , then for any , we have
It follows that ; that is, , for all . If we let , then by applying (A) we obtain .(ii) Given , we have and .

Given a two-step derivation on a trigroupoid , we denote its kernel by .

Proposition 13. Let be an fr-algebra. If is a two-step derivation on , then (i), ,(ii) implies that , ,(iii),for all .

Proof. (i) If we let in (24) and (25), respectively, then an , proving that , for any .(ii) If , then and for any , proving that , .(iii) If , then by Theorem 12, which shows that .

Proposition 14. Let be an fr-algebra. If is a two-step derivation on , then
for all .

Proof. If , then and hence
for all , which proves that .

Note that may not be computable unless the behavior of is specified, since for some .

Proposition 15. Let be an fr-algebra. If is a couplet of , then .

Proof. If is a couplet of and if , then for any . It follows from (6) that , proving that .

Let be a poset with minimal element 0. Define a binary operation “” on by
Then is a -algebra, called a standard -algebra inherited from the poset .

Proposition 16. Let be a standard -algebra. Let be an fr-algebra and let be a couplet of . If for some , then and .

Proof. Let . We claim that . Suppose that . Since is a fr-algebra, by applying (6), we obtain that is a contradiction. Since is a standard -algebra, we obtain . We claim that . If , then is a contradiction. It follows that , proving the proposition.

5. Classification of Linear Ranked Trigroupoids

Let be the real field and let be a ranked trigroupoid, where and are linear groupoids; that is, , , and for any , where (fixed).

Thus, if is a two-step derivation on , then for any , we have
and also in a similar manner we obtain

If for all , then (29) and (30) become
for any . If we let , in (31), then we obtain , . Hence, (31) reduces to
for all . If we let , and let , in (32) and (33), respectively, then we obtain
From this information, we obtain the following propositions.

Proposition 17. Let be the real field and let be a ranked trigroupoid, where are linear groupoids; that is, , , and for any , where (fixed). Let be a two-step derivation such that for all . (1)If and , then , , ;(2)If and , then , .

Proof. (i) If , then it follows from (34) that we obtain , . If , then we have and .(ii) If and , then and is arbitrary, and hence we obtain the result.

Proposition 18. Let be the real field and let be a ranked trigroupoid, where are linear groupoids; that is, , , and for any , where (fixed). Let be a two-step derivation such that for all :(i)if and , then , , ;(ii)if , , then , , ;(iii)if , , , then , , ;(iv)if , , , then , , ;(v)if , , , then , , .

In Propositions 17 and 18, we observe that there are 6 different types of linear ranked trigroupoids in the special case of for all , and most of them are classified by the properties of in .

6. Conclusion

The notion of two-step derivations is a generalization of derivations. This leads to the study of trigroupoids, and we explore some relations with several algebras, for example, -algebras, frame algebras, and so forth. The classification of linear ranked trigroupoids then explains a number of concrete algebraic structures with derivations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referee for valuable suggestions and help.